Prove \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}

Problem: 

Prove \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}

Answer: 

It is true that \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}